Need help trying to project shape onto cylinder...
I somewhat remade your scene, and was successful so I will contribute what I did. I think what may be causing you the biggest problem is the lack of geometry in your pyramid object. Try adding several loop cuts to support the boolean operation. 

Here is what I did:


Default cube, enter edit mode select top face and scale to 0 and scale as needed to create the pyramid object. (I also scaled the bottom face slightly).





Add several loop cuts in each direction with
Ctrl+R





In object mode, add a uv sphere, then go back into edit mode and
enable proportional editing by pressing O and select only
the top vertex.
Move the vertex on the z axis until you have the desired shape, and then scale to size.





Go into object mode and align the sphere object with the pryamid
object, and add a Boolean> Difference to the pryamid object, setting it's target to the sphere and delete the sphere from the scene or move to another layer.


After following these steps, also an edge split and subsurf modifier added- this is the result: